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Let \(\lambda\) be a weight. The crystals \(T_{\lambda}\), \(R_{\lambda}\), \(B_i\),
and \(C\) are important objects in the tensor category of crystals.
For example, the crystal \(T_0\) is the neutral object in this category; i.e.,
\(T_0 \otimes B \cong B \otimes T_0 \cong B\) for any crystal \(B\). We list
some other properties of these crystals:

The crystal \(T_{\lambda} \otimes B(\infty)\) is the crystal of the Verma
module with highest weight \(\lambda\), where \(\lambda\) is a dominant integral
weight.

Let \(u_{\infty}\) be the highest weight vector of \(B(\infty)\) and \(\lambda\)
be a dominant integral weight. There is an embedding of crystals \(B(\lambda)
\longrightarrow T_{\lambda} \otimes B(\infty)\) sending \(u_{\lambda} \mapsto
t_{\lambda} \otimes u_{\infty}\) which is not strict, but the embedding
\(B(\lambda) \longrightarrow C \otimes T_{\lambda} \otimes B(\infty)\) by
\(u_{\lambda} \mapsto c \otimes t_{\lambda} \otimes u_{\infty}\) is a strict
embedding.

Tensoring \(R_{\lambda}\) with a crystal \(B\) results in shifting the weights
of the vertices in \(B\) by \(\lambda\) and may also cut a subset out of the
original graph of \(B\). That is, \(\mathrm{wt}(r_{\lambda} \otimes b) =
\mathrm{wt}(b) + \lambda\), where \(b \in B\), provided \(r_{\lambda} \otimes
b \neq 0\). For example, the crystal graph of \(B(\lambda)\) is the same as
the crystal graph of \(R_{\lambda} \otimes B(\infty)\) generated from the
component \(r_{\lambda} \otimes u_{\infty}\).

There is also a dual version of this crystal given by
\(R^{\vee}_{\lambda} = \{ r^{\vee}_{\lambda} \}\) with the crystal
structure defined by

The crystal \(T_{\lambda}\) shifts the weights of the vertices in a crystal
\(B\) by \(\lambda\) when tensored with \(B\), but leaves the graph structure of
\(B\) unchanged. That is to say, for all \(b \in B\), we have \(\mathrm{wt}(b
\otimes t_{\lambda}) = \mathrm{wt}(b) + \lambda\).